A Cabling Conjecture for Knots in Lens Spaces

TL;DR

This paper constructs infinite families of hyperbolic knots in connected sums of lens spaces that admit lens space surgeries, generalizing the Cabling Conjecture to knots in lens spaces.

Contribution

It introduces a new construction method for knots in lens spaces with lens space surgeries and proposes a generalized Cabling Conjecture for these knots.

Findings

Constructed infinite families of knots with lens space surgeries

Unified previously known examples within this framework

Proposed a generalized Cabling Conjecture for knots in lens spaces

Abstract

Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space for all pairs of integers (r,s) except (0,0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz--Wu and Kang all fit this construction. As such, we propose a generalization of the Cabling Conjecture of Gonzales-Acuna--Short for knots in lens spaces.